Design of a 300 GHz Relativistic Gyrotron with an output Power of more Than 7 MW

Abstract

Calculations are presented for an electron-optical system that makes it possible to produce a helical electron beam with an energy of 250 keV, a current of 100 A, and a pitch factor of 1.1 for a 0.3 THz gyrotron with the operating mode of TE_33.2. Based on averaged stationary equations with a non-fixed field structure, the cavity profile is optimized and the possibility of obtaining an output power of about 8 MW with an electronic efficiency of more than 30% is demonstrated. Within the framework of particle-in-cell three-dimensional simulation, the processes of establishing oscillations are considered. Besides, it is shown that in the range of magnetic fields from 14.7 to 15.1 T, selective excitation of oscillations in the operating mode with a maximum power of about 7 MW is possible.

Appendix

The ANGEL (ANalyzer of a Gyrating Electrons) software package was developed on the basis of well-established algorithms for the analysis of the electron-optical systems based on the methods of current tubes and discrete sources [61, 62].

Numerical simulation of helical electron beams is carried out by means of an approximate solution of the system of the equations of motion in a cylindrical coordinate system (r,φ,z), assuming the axial symmetry of the system (${\partial \mathord{\left/ {\vphantom {\partial {\partial \varphi }}} \right. \kern-0pt} {\partial \varphi }} = 0$):

$$\frac{\partial (\gamma \bf{v})}{\partial t}=\eta \left(\bf{E}+\bf{v}\times \bf{B}\right),\hspace{1em}\frac{\partial \bf{r}}{\partial t}=\bf{v},\hspace{1em}\bf{r}\left|{}_{t=0}\right.={\bf{r}}_{\hspace{0.05em}0}\hspace{1em}\bf{v}\left|{}_{t=0}\right.={\bf{v}}_{0},$$

(A1)

where r, v are the vector of the coordinate and velocity of the particles, r₀, v₀ are their initial values. In turn, the electric field E is calculated based on the Poisson equation in combination with the continuity equation [63]:

$$\bf{E}=-\nabla U,\hspace{0.33em}\frac{{\partial }^{2}U}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+\frac{{\partial }^{2}U}{\partial {z}^{2}}=-\rho ,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}U\left|{}_{Boundary}\right.={U}_{B},\hspace{0.33em}{\left.\frac{\partial U}{\partial r}\right|}_{r=0}=0,\hspace{0.33em}\hspace{0.33em}\text{div }\bf{j}=0,$$

(A2)

where v is the electron velocity, η is the specific particle charge, γ is the relativistic gamma factor, ρ and j = ρv are the space charge density and electron beam current density, respectively. The potential U_B at the boundary corresponds to the potential of the EOS electrodes.

The trajectory analysis of the electron flow is performed by the current tubes method, and the solution of the system of self-consistent equations is carried out by the method of successive iterations. At the initial (zero) iteration, the space charge density is assumed to be zero and the solution of the Laplace equation is found under given boundary conditions. The flow of particles emitted from the emitter is divided into N current tubes (i.e. each of them carries the current I_k, k = 1..N, which remains during the movement). The space charge introduced by k-th tube into one cell of the grid is Q_m,k = I_k ∆t, where ∆t is the time spent by the particle within the cell, the total charge of the cell Q_m is obtained by summing the charges Q_m,k over all current tubes that have passed through the given cell. Integration of the equations of motion allows to determine the trajectories of each current tube and the distribution of the space charge density. At the next iterations, the electric field is found from the solution of the Poisson equation with the density ρ calculated at the previous iteration, and then the electronic trajectories are calculated again.

The magnetic field is assumed to be external, its induction B is created by a system of solenoids with a rectangular longitudinal section. The magnetic induction of an azimuthally symmetric solenoid bounded by longitudinal coordinates z₁ < z₂ and radial coordinates r₁ < r₂ is determined by the azimuthal component of the vector potential A:

$${B}_{r}=-\frac{\partial {A}_{\varphi }}{\partial z},{\hspace{0.33em}}{B}_{z}=\frac{\partial {A}_{\varphi }}{\partial r}+\frac{{A}_{\varphi }}{r},{A}_{\varphi }\left(z,r\right)=\underset{0}{\overset{\pi }{\int }}\underset{{r}_{1}}{\overset{{r}_{2}}{\int }}\underset{{z}_{1}}{\overset{{z}_{2}}{\int }}\frac{\widetilde{r}{\text{cos}}\varphi }{\sqrt{{\left(\widetilde{z}-z\right)}^{2}+{\widetilde{r}}^{2}+{r}^{2}-2r{\hspace{0.05em}}\widetilde{r}{\text{cos}}\varphi }}d\varphi d\widetilde{r}d\widetilde{z}$$

(A3)

The field calculation is optimized by analytical calculation of integrals over the radial and longitudinal coordinates, which significantly increases the accuracy and speed of calculation [64].

The calculation of the electric field is carried out by the method of discrete sources, the use of which is effective due to the small volume occupied by the electron flow relative to the total volume of the interelectrode space. The essence of the method is reduced to the representation of the scalar potential U by a linear combination of discrete sources – «virtual charges» Q_i, located along the boundary at points (ρ_i,ζ_i), spaced from the surface of the electrodes «into» the metal at a distance of the order of a step:

$$\begin{gathered} U = \sum\limits_{i = 1}^{N} {Q_{i} G\left( {r,z,\rho_{i} ,\zeta_{i} } \right)} + U_{S} ,\;G\left( {r,z,\rho_{i} ,\zeta_{i} } \right) = \frac{1}{{2\pi^{2} \varepsilon_{0} }}\frac{{{\kern 1pt} F(k)}}{\sqrt \chi }, \hfill \\ k^{2} = \frac{{4r\rho_{i} }}{\chi },\;\chi = (r + \rho_{i} )^{2} + (z - \zeta_{i} )^{2} ,\;F(k) = \int\limits_{0}^{{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} {\frac{d\beta }{{\sqrt {1 - k^{2} \sin^{2} \beta } }}} , \hfill \\ \end{gathered}$$

(A4)

where U_s is the potential generated by the space charge of the beam at the previous iteration. At the zero iteration, U_s = 0 is assumed, and at subsequent iterations, it is found using the following formula:

$${U}_{s}=\sum_{m=1}^{M}{Q}_{m}G\left(r,z,{\rho }_{m},{\zeta }_{m}\right),$$

(A5)

with the same Green’s function G as when finding the potential at the boundary, M is the total number of grid cells to consider the space charge of the electron beam.

The values of the auxiliary charges Q_i are found from the system of linear algebraic equations obtained by checking the boundary condition (the potential is equal to the given U_B) at several discrete points of the boundary, located with a certain step on the surface of the electrodes and called collocation points:

$$\sum\limits_{i = 1}^{N} {Q_{i} G\left( {r_{j} ,z_{j} ,\rho_{i} ,\zeta_{i} } \right) = U_{B} \left( {r_{j} ,z_{j} } \right)} - U_{S} \left( {r_{j} ,z_{j} } \right),\;\;j = 1..N.$$

(A6)

The derivatives of the potential with respect to directions are taken analytically. To optimize the calculating time, a grid of "enlarged charges" is used. To integrate the equation of motion, the 4^th order Runge–Kutta method is used i.e. usual and with a modification of Merson, which makes it possible to control the accuracy at each step.